Powell’s Pi Paradox: the genius 14th century Indian solution

above. For example, the fourth term here is just the difference between the fourth and third elements of the above sequence. good? Blue minus green, the first three terms cancel. there. There, what remains is -4/7, the fourth term of our series at the bottom. The same goes for all the other terms. There, blue minus green equals orange. Blue minus green equals orange. Blue is the same as orange. Now let's add the correction conditions. This gives a new series of numbers. Now, which chain corresponds to this new sequence? easy. As before, its terms are simply the differences between successive elements of the sequence.

As if the fourth term here is just the difference between the fourth element and the third element. Blue minus green, well, the same conditions above cancel each other out. There and there. So, the fourth term of the new series is this. Now, when an expert thinks a little about the general equation for the ninth difference, he will notice that this term can be rewritten in this very nice short form. It's not obvious but at the same time it's just a bit of algebra. Then the other terms are calculated in the same way. There and there and there.

In the same way, all correction terms can be converted to new infinite series formulas for Pi. However, the first two appear only in those ancient Indian manuscripts. Because? Perhaps because only in these first two can the terms of the new series be written in a surprisingly simple and beautiful way. Furthermore, in those ancient Indian palm leaf manuscripts, not all of these formulas were written in the concise mathematical language we use today. There they were all, believe it or not, all expressed in verse! And I think you thought twice before putting something ugly in the verse :) Anyway, that's all for today.

I leave you with a detailed look at a page from one of these ancient manuscripts. Let's see if you can spot any x :)